When \(d=2\) , it is easy to see that \({\bf V}_N^k\)  is the same space of \({\bf V}_h^k\)  in Section REF  with uniform partition and \(N_x = N_y =2^N\) .
On the other hand, a standard choice of sparse grid space [1]}, [2]} is
\(\hat{{\bf V}}_N^k=\bigoplus _{\begin{array}{c} |\mathbf {l}|_1 \le N\\\mathbf {l}\in \mathbb {N}_0^d\end{array}}\mathbf {W}_\mathbf {l}^k \subset {\bf V}_N^k.\) 
